# Estimation of Input Signal to Obtain the Desired Output Signal for an Unknown Filter

Suppose $h(n)$ is a finite impulse response which is unknown. We can feed any input signal $x(n)$ into the system and observe the corresponding output signal $y(n)$.

From this, is it possible to estimate the input signal $x^*(n)$ which when passed through the filter gives the desired output signal $y^*(n)$ ?

I think a Least Mean Squares filter could be used here. Could anyone point out how one proceeds doing this ?

Thanks !

• Short pointer: the technique that you're looking for is called deconvolution (or, for your case specifically, you're trying to do system identification). You are correct in that LMS is one way to tackle the problem. – Jason R Sep 25 '13 at 15:52

https://workspace.imperial.ac.uk/earthscienceandengineering/Public/lecture%20handout%2019Jan09.pdf

In the most simple case, just to give intuition about the problem, it is really easy.

In the Frequency Domain:

$${Y}^{\ast} \left( \omega \right) = H \left( \omega \right) {X}^{\ast} \left( \omega \right) \Rightarrow {X}^{\ast} \left( \omega \right) = \frac{ {Y}^{\ast} \left( \omega \right) }{ H \left( \omega \right) }$$

Since ${Y}^{\ast} \left( \omega \right)$ is known all needed is $H \left( \omega \right)$.

Yet since we have access to a black box of $H \left( \omega \right)$ we can set input of a known signal $X \left( \omega \right)$ and have the output $Y \left( \omega \right)$ which will give us $H \left( \omega \right)$:

$$Y \left( \omega \right) = H \left( \omega \right) X \left( \omega \right) \Rightarrow H \left( \omega \right) = \frac{ Y \left( \omega \right) }{ X \left( \omega \right) }$$

As others pointed out it is called Deconvolution and as you mentioned it can be done using Least Mean Square (LMS) Filter.